Assembly Sequence Influence on Geometric Deviations of

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Assembly Sequence Influence on Geometric Deviationsof Compliant Parts

Mathieu Mounaud, François Thiebaut, Pierre Bourdet, Hugo Falgarone,Nicolas Chevassus

To cite this version:Mathieu Mounaud, François Thiebaut, Pierre Bourdet, Hugo Falgarone, Nicolas Chevassus. AssemblySequence Influence on Geometric Deviations of Compliant Parts. International Journal of ProductionResearch, Taylor & Francis, 2010, pp.1. �10.1080/00207540903460240�. �hal-00564093�

https://hal.archives-ouvertes.fr/hal-00564093

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Assembly Sequence Influence on Geometric Deviations of

Compliant Parts

Journal: International Journal of Production Research

Manuscript ID: TPRS-2009-IJPR-0506

Manuscript Type: Original Manuscript

Date Submitted by the Author:

01-Jun-2009

Complete List of Authors: Mounaud, Mathieu; LURPA Thiebaut, François; LURPA-ENS Cachan Bourdet, Pierre; LURPA-ENS Cachan Falgarone, Hugo; EADS-IW Chevassus, Nicolas; EADS-IW

Keywords: TOLERANCE ANALYSIS, ASSEMBLY PLANNING, SEQUENCING

Keywords (user):

http://mc.manuscriptcentral.com/tprs Email: [email protected]

International Journal of Production Research

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Assembly Sequence Influence on Geometric Deviations Propagation

of Compliant Parts

M. Mounaud1**

, F. Thiébaut1,2

, P. Bourdet2, H. Falgarone

3, N. Chevassus

3

1LURPA, ENS de Cachan, 61, Avenue du Président Wilson, 94235 Cachan Cedex,

France 2IUT Cachan, 9, Avenue de la division Leclerc, 94235 Cachan Cedex, France

3EADS-IW, 12, rue Pasteur, 92150 Suresnes, France

(Received 01 June 2009;)

This paper presents a non-rigid part variation simulation method for fulfilling functionnal

requirements on compliant assemblies. This method is based on the propagation of

different geometrical deviations (manufacturing and assembly process defects) using the

Method of Influence Coefficient. Tolerance analysis of compliant assemblies is also

achieved very early in the design stage. As a consequence, designers and manufacturing

engineers can efficiently analyse the assembly design principles both in terms of installed

stresses and geometric variation clearance. They can also set optimised' sequences that

enable to get rid of geometric variations.

Keywords: Tolerancing Analysis, compliant parts, assembly sequencing

Introduction

To reduce energy consumption, automotive and aeronautics industry are concerned

with decreasing the weight of their assemblies while using more and more systems to

improve their products -avionics, electronics, hydraulics or mechanical systems, for

instance. The important developments in computer engineering in the last three

decades allow these industries to model more and more components during the design

stage. The complexity may be defined in terms of material or mechanical behaviour,

geometry. It also involves manufacturing overconstrained assemblies taking into

account part compliance.

The reduction in time development leads these industries to simulate the

behaviour of their assembly as accurately as possible and at the earliest stage, in order

to minimize the need for physical tests and to improve their producibility. A way to

achieve this aim is the use of variation simulation tools as soon as possible.

* Corresponding Author. Email : [email protected]

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In early design, the nominal dimensions of parts are decided and tolerances

and locating schemes are assigned. Assemblies with such parts have to respect some

functional requirements like geometric clearance or installed stresses in the

assemblies’ joints. At any time, it is important to make sure that the functional

requirements of a complex assembly are satisfied. Söderberg [1] explains that

variation in a geometrical key characteristic of an assembly depends on three different

sources: component variation, assembly variation and design concept as shown in

Figure 1. Mechanisms in automotive or aircraft industries are composed of compliant

parts with deviations from nominal shapes. Moreover such parts may be jointed

together with an important number of joints which overconstrain the considered

assembly. In this context, the industrial need for variation simulation tools used in the

design stage increases quickly, while the development and pre-production time

decreases.

Please Insert Figure 1

There are different ways to manufacture an over-constrained assembly of

components whose geometry has defects:

• Components should have restrictive tolerances. This implies important costs,

which makes the industry less competitive.

• Some kinematic adjustments between parts may be used to compensate for

assembly variation. Functional requirements may not be expected.

• Components may be compliant to compensate for parts’ defects.

• Finding the best equilibrium between the last two points.

To help designers analyze their products, it is thus important to provide them

with tools to perform the best and fastest choices; more particularly to predict how the

design and set of tolerances they assign to assembly compliant parts affect the

product's functional requirements.

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Scope of the paper

This paper presents a method based on different existing approaches to

analyze an assembly composed of compliant parts. Used in the design stage, the

proposed method allows engineers to verify if functional requirements of final

assembly are fulfilled. The proposed method relies on two major axes which are

tolerance analysis and compliant assembly variation.

After a description of recent advances, the simulation tools that are

used are developed. To illustrate this theory, an aeronautic hydraulic system assembly

is presented in section four as a case study. The last section discusses the results and

concludes the paper.

Literature review

Over the past years, many different approaches to Tolerance analysis and Compliant

Assembly Variation have been developed.

Many researchers have worked on tolerance analysis and developed different

models, Wirtz (1998) studies, for example, vectorial tolerancing whereas Pillet et al.

(2005) worked on inertial tolerancing. Torsor and Jacobian approaches were

respectively developed by Ballot et al. (1995) and Laperriere et al. (2002), just to

name a few. Laperriere and Kabore (2001) have already worked on tolerance

synthesis with Monte Carlo simulation, but most of these works deal with tolerance

analysis of rigid-body components, which is far from the exposed context.

Tolerance analysis for compliant parts consists in predicting the amount of

misalignment that will occur between parts, but also enables to predict the amount of

deformation of parts and stresses in joints. Many research works have focused on

predicting dimensional variation on sheet metal assembly. Most of the developed

methods rely on Finite Element Analysis (FEA). Chang and Gossard (1997) simulated

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the assembly and measurement processes with part stiffness matrix to find the final

variation of an assembly. Liu and Hu (1997) proposed a model to analyze the effect of

part geometrical deviations and assembly springback on assembly variation. The

method based on sensitivity matrix, called method of influence coefficient, combines

FEA and Monte Carlo simulation to reduce computational costs. Ceglarek and Shi

(1997) presented a Beam-based model for tolerance analysis for sheet metal

assemblies. Merkley (1998) proposed a tolerance analysis of compliant parts which

accounts for the part covariance in conjunction with the part stiffness matrix. Cid et

al. (2005) established an extension of 3D link chains to provide symbolic equations of

the assembly by integration of compliance and actual deviation from the nominal

shape of the considered part. Breteau et al. (2007) developed this approach for the

measurement of compliant parts. Most of these works, based on the study of one

compliant part, have been extended to mechanisms composed of several parts and/or

to the integration of different parameters influencing assembly variation.

Hu et al. (2006) summarized recent developments in simulation models for

compliant assemblies and presented different applications. Based on assembly state

space models and multi-station levels for assembly variation analysis, Camelio et al.

(2003) developed these applications for variation analysis, robust design, tolerance

allocation and variation reduction. The state space model has been extended by Loose

et al. (2009) with the development of an analytical derivation to describe features

deviations controlled by GD&T characteristics. For tolerance analysis some aspects

such as assembly sequence, contact modelling, we refer to Dahlström et al. (2005),

and fixture layout, referring to Huang et al. (2007), need to be incorporated to find a

correlation between simulated variations and actual ones. Söderberg et al. (2007)

presented a tool for non-rigid variation simulation and visualization based on

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sensitivity matrix and more particularly on the unit influence coefficient method. This

paper proposes to extend this approach by taking into account relevant aspects

exposed before such as assembly sequence, fixture defects and compliance,

manufacturing variations in order to check if functional requirements of a compliant

hydraulic assembly, we refer to Mounaud et al. (2007), are satisfied. Wei (2001) used

a similar approach to the one proposed to predict the fulfilment of functional

requirements of such assemblies by taking into account geometrical variations in

tubes and in structures in order to evaluate the cost of different designs. But the

proposed models are restricted to tube compliance with no influence of the assembly

sequence.

Description of the proposed method

Part compliance enables assembly but it also causes considerable variations that

influence the final geometry of the assembly. As compliant parts may be linked to

others parts (compliant or not) with important number of joints it over-constrains the

considered mechanism.

To be functional, assemblies of compliant parts need to fulfil some functional

requirements (see equations 1 and 2) which are usually residual stresses “Fi” of the

assembly in each joint “i” and geometrical requirements “Ui” on specific points of the

assembly.

i maxF F< (1)

i maxU U< (2)

To predict the fulfilment of the above functional requirements the proposed

method is carried out according to the seven following steps detailed in the current

section:

(1) Definition of each part's deviations.

(2) Definition of the mechanical behaviour of each part.

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(3) Definition of the mechanical behaviour of the assembly

(4) Calculation of the Influence matrix based on the unit displacement method and

part stiffness matrix from Finite Element Analysis

(5) Identification of the most robust fixturing scheme for part localization.

(6) Simulation of an assembly sequence

(7) Results

Actual Geometry of components

Considering Tolerance Analysis and assembly simulation, the actual geometry of

parts is supposed to be known thanks to admissible geometries described through

geometric specifications or indirectly by the way of manufacturing parameters.

Whatever the definition of admissible geometry, a representative set of geometry

instances is supposed to be known for each part of the assembly.

Let us suppose a frame is attached to each component of the assembly. For

each point Ci of the nominal component, the manufacturing deviation dM (Ci) from

the nominal model to the component is known and expressed relatively to the

component frame. The actual position of a point � iC of the component is known by

the equation 3:

�i i M iC C d (C )= + (3)

Definition of mechanical behaviour of each component

Tolerance analysis is applied on assemblies that mix rigid components and compliant

components.

• Considering compliant components, geometric and mechanical behaviour are

linked through a rigidity matrix since the following hypotheses are used:

- the deformation of the component remains linear,

- material behaviour is isotropic,

- small displacements are considered.

Let the deformation of the component be expressed relatively to the associated

frame. Let Ci be a nominal point of a component C, the position of the point that

results in a small displacement of the component frame and of the component

deformation is given by the equation 4.

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[ ]*i i i

R TC C U(C )

= +

0 0 0 1 (4)

where R represents the small rotation of the frame, T represents the translation

of the origin of the frame and U represents the displacement of the Ci point due to the

component deformation (as shown in figure 2).

Please insert Figure 2

The displacement of the Ci point due to the component deformation is

implicitly known by the linear relations between exterior mechanical solicitations F

and displacements U:

F K.U= (5)

where K is the matrix rigidity of the component.

• Considering rigid components, the geometric behaviour is defined through screw

theory as far as geometry is concerned and static as far as mechanical behaviour is

concerned.

Let Ci be a nominal point of a component C, the position of the point that

results in a small displacement Ci* of the component frame is given by the equation 6

[ ]*i i

R TC C

=

0 0 0 1 (6)

where R represents the small rotation of the frame and T represents the

translation of the origin of the frame.

Definition of mechanical behaviour of the assembly

The behaviour of the assembly is known through the behaviour of each component

and the links between components. Using a finite element method, the behaviour of

links between components is known as the boundary conditions.

A link behaviour is defined by equations that represent the interface between

two components expressed in terms of transmissible mechanical solicitation (contact

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is insured) and degrees of freedom (mechanical solicitation is null) as it is explained

by Mounaud et al. (2007).

For each interface point, the equations result from:

• The mechanical equilibrium of the interface.

• The continuity of the displacement when contact is effective.

• The absence of mechanical solicitation when a degree of freedom exists.

• The stiffness of the interface when the interface is elastic.

As the values of the mechanical deviations are supposed to be known, the

solution to the set of equations provides the values of the unknown in terms of

displacements and mechanical solicitations.

Calculation of Influence matrix based on the unit displacement method and part

stiffness matrix from Finite Element Analysis

A way of performing the simulation faster than solving the set of equations for each

instance of geometry is to establish a linear relationship between part deviations and

assembly spring-back deviations by using the Method of Influence Coefficient

developed by Liu and Hu (1997). This method can be applied as part deviations from

the nominal geometry are considered small and the material properties remain in the

linear range.

In this method, a unit displacement is applied at the i-th source of variation ,

i=1 to 6*p, where p is the number of links. Then the set of equations is used to

calculate the response fi under that unit displacement.fi represents a vector of

influence coefficients associated to the i-th source of variation. If the response is

recorded for all the sources of variation, the matrix of influence coefficients S is

obtained. As the mechanistic problem is considered a linear one, using the property of

superposition of linear systems, the total imposed displacements ud of the tube can be

expressed as a linear combination of unit displacements used for establishing matrix

S:

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tot df S * u= (7)

Consequently, all the installed forces ftot (at each joint of the tube) can be

calculated for all the configurations of the actual pipe geometry obtained with Monte

Carlo simulations. As the assemblies studied are over-constrained, it is important to

find an optimal datum choice to localize the actual tube’s geometry.

Identification of robust fixturing scheme

To simulate the influence of the assembly sequence on the functional requirements the

assembly has to satisfy, the first step is to find the most robust fixturing scheme. That

consists in finding the best set of joints to minimize the geometrical deviations of the

assembly. Each joint of a compliant part with others can be described as a

combination of elementary joints i.e. point contact locator characterized by both its

own point contact “Pi” and its outward unit normal vector at contact point nij. ‘j’

represents the elementary joint composing the i-th joint in case of same point contact.

To illustrate this, if a joint is considered a planar joint, it will be defined as a

combination of 3 point locators in the same plane. For each point locator it is possible

to define plückerian coordinates with equation 8:

{ }i

ijpi j

i ij P

nT

OP n

= ×

(8)

where O is a fixed point.

Deterministic localization is a fundamental requirement indicating that the part

cannot make infinitesimal motion without losing contact with at least one locator.

Some authors like Zhaoqing et al. (2008) use genetic algorithms to find the optimal

fixture layout which sensitivity of product variation to fixture errors is minimized.

The method proposed by Wang et al. (2001) allows to achieve this requirement only

if a matrix called locator matrix, locM defined by equation 9, has its full rank of 6.

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The proposed locator matrix in this paper is composed of the plückerian coordinates

of 6 elementary joints among all the possible elementary joints.


A classical 3-2-1 fixture scheme, represented on Figure 3, has its locator

matrix with the following form:

{ } { } { } { } { } { }( )loc P P P P P PM T T T T T T= 1 2 3 4 5 61 1 11 1 1 (9)

If there is a choice of 6 locators among N, at most )!!*(

!

6N6

NC

6N −−−−

==== possible

combinations to achieve this scheme exist. Maximizing the determinant of the locator

matrix is a suitable criterion for accurate localization as shown by Wang and

Pelinescu (2001) i.e. ( )locmax det M . As the method may be very time-consuming, a

reduction of the combination can be achieved by avoiding wrong cases of

localization. For example, if a fourth locator exists on the principal plan of figure 3,

there is a choice of 6 among 7 locators to make. Thus, solutions with 4 locators on the

principal plan which over-constrain the layout of the part can be ruled out before the

combinatory method.

Simulation of assembly sequence

A simulation of assembly sequence predicts more correlated results with physical

tests than the FEA, based on the minimization energy principle detailed in section 3.2.

Camelio et al. (2003) considered the assembly sequence as a multi-station assembly if

it is assumed that a step of the assembly is represented as a station.

At each stage of the assembly, the equation (10) is equivalent to the following

one:

[ ]u

K I * [ ]f

− =

0 (10)

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Formally, force or displacement values can be imposed on every point leading

to the boundary conditions. Writing all the boundary conditions leads to the equation

11:

BC

K I u*

SBoundaryConditions f

− =

0 (11)

Considering that geometrical deviations are small and do not affect stiffness

matrix coefficients, only the boundary conditions change while the assembly

sequence proceeds. The propagation model of manufacturing variations, detailed in

section 3.1, describes the gaps that must be compensated for when assembling the

actual part on its joints. Only joints concerned at the current step of the assembly

sequence are assembled. At each stage of the assembly sequence, boundary conditions

have to meet the requirement set by the distance function defined below:

P f P fM M ef ef RBMd d d d d d d= + + + + +0 (12)

Where the different terms of the above equation are:

• d: actual distance between part and fixture

• d0: nominal distance between part and fixture (=0 in case of assembly)

• dMP: Manufacturing deviation of the part

• dMf: Manufacturing deviation of the fixture

• defp: deformation of the part

• deff: deformation of the fixture

• dRBM: rigid body motion if exists

The different distances can also be illustrated in the following Figure 4:


Although illustrations have been made in 1-D, as distances are defined as

spatial vectors, 3-D problems can be solved thanks to this representation. As the

actual distance between the part and the fixture has to be null to make an assembly,

the boundary conditions can be solved. At a step of the assembly sequence, gaps of

the considered joints thus constitute the boundary conditions the mechanistic problem

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has to take into account. Consequently, this displaces all the part geometry of the part

at the beginning while the mechanism can have a rigid body motion. Once the part is

totally constrained, from a kinematical point of view, its geometry deforms only under

the applied displacements. The solution found at the i-th stage represents the gaps that

need to be closed at the i+1-th stage. The method is iteratively going on until the end

of the assembly sequence.

Selection and representation of results

The simulation of an assembly sequence is performed for instances of manufacturing

deviations. The Monte Carlo method is used so that the results are representative of

the real behaviour of the assembly. Synthetic representation has been chosen because

the amount of available data is too important. Three representations of results are

proposed. The first one provides an estimation of how the characteristics are

respected, as shown in table 1.

Please Insert table 1

The global estimation of the conformity of the assemblies allows designers to

validate the design or to modify the design if the rate of conformity in not acceptable.

When this situation occurs, a two-dimensional or three-dimensional representation of

the results of simulations is proposed.

Figure 5a illustrates a two dimensional representation of force distribution, the

choice of the local projection plane is free. On the figure, the circle represents the

admissible limit in term of force. Each point corresponds to the projection of a

simulation result in the plane.

Please insert Figure 5a and Figure 5b

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As a complement to the previous representation, the distribution of any

evaluated data is proposed and illustrated using a histogram (Figure 5b).

Case Study

The proposed method is illustrated trough an aeronautic example which

consists in assembling a hydraulic tube on its brackets as shown on figure 6. The

metallic tube must satisfy the functional requirements in terms of geometrical

deviations and limited effort in each joint. It is assumed that tube and brackets are

compliant. The tube is fixed with 6 brackets on its route and 2 brackets on its

extremities. Moreover, the joints are called Controlled Check Points (CCP). To

position the tube, joints are thus considered as rotational joints for both ends and as

sphere-cylinders in other cases. The main geometrical characteristics of the tube and

its material properties are detailed in the following table 2:


Please Insert Figure 6 and table 3 (if possible)

The objective of the study is to simulate the assembly of the tube on a

structure. The deviations on structure have already been evaluated and the machining

process of the tube is known. At this stage of the design, the designer has to chose the

technology of the brackets, to impose the assembly planning and to verify that the

choices permit to respect the functional requirements on the assembly.

For a given choice of bracket technology, we propose to simulate the assembly

of the pipe on the structure without taking into account the influence of assembly

planning on specification fulfilments, This simulation is not time consuming and

provides pertinent indicators about the assembly feasibility. Once the assembly

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feasibility is assumed, a more detailed study is performed to verify that functional

requirements are respected while taking into account the influence of assembly

planning.

We now propose to detail the seven steps of the method applied to the case

study.

Definition of each part deviations

As mentioned above, deviations on the structure have already been evaluated. This

previous study provides the dispersions of the positions of locating holes of the

brackets relative to their nominal position. Locating holes appear to be centred on

nominal position and the deviations respect a normal distribution. The structure is the

base of the assembly, its frame (O,X,Y,Z) is the reference of the assembly.

Please insert table 4

In order to minimize both the mass and the machining cost of the six brackets,

a unique technology has been chosen. The deviation of the position of the bracket

centre, relative to its base, is defined in the frame associated to the bracket ( refer to

table 5). Bracket deviations can thus be expressed in the frame of the structure.

Please Insert Figure 7 and table 5

Concerning the pipe, the manufacturing process (bending process) is known,

and the capability of the bending process is identified. As shown in Mounaud et al.

(2007), pipe geometry is classically represented as a sequence of elementary bending

operations made with a forming die which radius is called Rf. These operations are

characterized by L-R-A manufacturing coordinates, where L,R and A stand for

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Length between bends, Rotation between two successive bends and bent Angle,

respectively, as shown in Figure 8.


As a tube is made from a sequence of elementary operations, it is assumed that

the geometry can be considered as a sequence of successive straight and circular

elements. So local frames Ri (Pi,Xi,Yi,Zi) can be associated with elementary elements

i. Due to the manufacturing process, these coordinates (Li, Ri and Ai) are made with

deviations (dLi, dRi and dAi respectively) limited by the process capabilities dL, dR

and dA represented by the following constraint relations (equation 13, 14 and 15) for

each actual elementary bending operation iL~

( iii dLLL~

+= ), iR~

( iii dRRR~

+= ) and

iA~

( iii dAAA~

+= ):

i i idL dL

L L L− ≤ ≤ +2 2

% (13)

i i idR dR

R R R− ≤ ≤ +2 2

% (14)

i i idA dA

A A A− ≤ ≤ +2 2

% (15)

Figure 9 represents the distribution of the dL process capability. dR and dA

are assumed having similar respresentations.


The actual geometry of a pipe can easily be obtained by representing each

elementary operation and its own defect with the Homogenous Transformation Matrix

for L, R and A respectively:

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i

R

LM

=

1000

0100

L~

010

0001

i

i

i

R

RM

−=

1000

0)R~

cos(0)R~

sin(

0010

0)R~

sin(0)R~

cos(

ii

ii

i

i

R

AM

−−

=

1000

0100

)A~

sin(*R0)A~

cos()A~

sin(

)]A~

cos(1[*R0)A~

sin()A~

cos(

ifiii

ifiii

Rfi is the actual radius which integrates spring-back phenomenon as indicated

in Lou and Stelson (2001). Thus, from the bend plan, the geometry of the actual tube

may be generated in 3D-space by combining the effect of multiple bends through a

series of matrix multiplications. As a consequence, in a global Cartesian coordinate

system, the relative position (Pi) of the different specific points (Piloc in the local

coordinate system of the elementary element i) of the tube is obtained with the

following equation 16:

k i

k k k iloc

k

Pi ( ML * MR * MA )* P

=

=

= ∏1

(16)

The bending process is characterized by the following values: 1mm, 0.1° and

1° for dL, dR and dA capabilities, respectively.

Definition of the mechanical behaviour of each part

The structure of the assembly is supposed to be rigid and the manufacturing

deviations are known. As the structure is the base of the assembly, its frame (O, X, Y,

Z) is the reference of the assembly (cf. Figure 7).

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The brackets 2 to 7 are considered as compliant with a linear model of

deformation (5N.mm along axis in translation and 5Nm.rad along axis in rotation).

The elementary stiffness matrix of a bracket is given equation 17 and 18:

i i i i

j

CCP ,X ,Y ,Z

K

=

5 0 0

0 5 0

0 0 5

(17)

i i i i

j

CCP ,X ,Y ,Z

C

=

5 0 0

0 5 0

0 0 5

(18)

Moreover, it is assumed that extremities are considered as rigid CCP

compared to brackets. So they are modelled with the elementary matrices in equation

17 and equation 18 where values of stiffness are respectively equal to 100 N.mm and

100 Nm.Rad. Considering the tube, the Finite Element Modelling used relies on a

classical strength of material approach based on Bresses’ equations. The model used

is detailed explicitly in Mounaud et al. (2007). The simulation assumes that the tube

deformation remains linear, that material behaviour is isotropic and that only small

displacements are considered. The result of the Finite Element Modelling is the

stiffness matrix of the tube.

Definition of mechanical behaviour of the assembly

The boundary conditions that permit to link the local behaviours of the components

are now presented.

• For each interface between the structure and the brackets, the boundary condition

is the continuity of displacement since contact is assured. Let Si be the point of the

structure and Bi be the corresponding point of the bracket, then: �

i iPosition(B ) Position(S )= (19)

These equations 19 allow us to calculate the small displacement of the

considered bracket frame since the positioning of the bracket is equally constrained.

• For each interface point between the bracket and the pipe, the position of the

actual bracket point is known since the small displacement of the bracket frame

and the manufacturing deviation are known. The geometry of the tube is also

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available and then it is possible to calculate the value of the distance function

presented in 3.6.

The technology of the bracket allows us to consider the link between the tube

and the bracket as a sphere-cylinder link while the bracket is not clamped, and as a

complete link when the bracket is clamped.

The boundary conditions associated to a complete link are the continuity of the

displacement in all directions.

The boundary conditions associated to a sphere-cylinder link are:

• The continuity of the displacement along Yi and Zi directions.

• The absence of mechanical solicitation along axis where degrees of freedom exist

(Xi direction in our case).

For each interface point, the remaining equations result from the mechanical

equilibrium of the interface.

Calculation of Influence matrix based on the unit displacement method and part

stiffness matrix from Finite Element Analysis

As the values of the mechanical deviations are supposed to be known, the solution to

the whole set of equations provides the values of the unknown in terms of

displacements and mechanical solicitations.

Using the method of the influence matrix, detailed in section 3.4, current assembly

positions can be assessed and then the joint forces can be calculated at each step of the

assembly sequence.

Identification of the most robust fixturing scheme for part localization.

The only component, for which the identification of the most robust fixturing scheme

is needed, is the tube. The application of the method developed by Wang and

Pelinescu (2001) to find the best fixturing scheme indicates that the tube is positioned

with CCP 3, 6 and 7. The latter are called “positioning CCP” and the others,

“clamping CCP”.

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Simulation of an assembly sequence

This step of the method only depends on the assembly sequences of clamping. For

this case study, three sequences are studied as shown in table 6.

Please insert table 6

Results

For each sequence, 5000 simulations are performed using Monte Carlo method.

The indicators that result from sequence 1 simulations are presented in the

table and indicate a global satisfaction of functional requirements, except for force

criterion at point 1 and 8 and for displacements at point 2.

Please Insert Table 7

Local representations of the forces and displacements at problematic points are

presented in figure 10.

Please insert figure 10

From the local representations, we can conclude the global orientations of

forces and displacements concerning points 1 and 2 and a homogeneous repartition

concerning point 8.

As the results of simulations are simplified, a more detailed analysis is

proposed in order to quantify the influence of assembly planning. Two alternative

assembly sequences are studied and the global results are presented in table 8 and 9.



The global results in sequence 2 (Clamping Sequence CCP 2 6 7 5 4 3) show a

potential improvement in the fulfilment of functional requirements in terms of forces.

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Local results concerning efforts at point 1 and 8 and displacements at point 2

are proposed for this sequence analysis in figure 11.


The yield rate for the force requirement criterion is lower, so this assembly

could be accepted concerning forces. The remaining problem concerns the

displacements at point 2. The influence of assembly planning is negligible and the

yield rate is not acceptable.

At this stage of the design, bracket technology may still change. As the

designer has effort and displacement repartitions at his disposal, he can take

advantage of them to suggest a modification. Concerning point 2 where the problem

occurs, we note in figure 12 that the range of displacement along UZ2 axis exceeds the

limit, while the range of force along UZ2 axis does not reach the limit.


An increase in the stiffness of the bracket along Z2 direction may increase

forces and reduce displacements to produce an acceptable assembly design.

Conclusion and future work

Taking into account the assembly sequence of components in an aeronautic assembly

contributes to the anticipation of the geometrical part deviations to verify that the

deformed geometry meets functional requirements. The purpose of this paper is to

present a model for the influence of assembly sequence which integrates

manufacturing deviations for the fulfilment of product key characteristics. Part

manufacturing deviations have been taken into account. Due to the assembly sequence

of the structure parts, CCP positions present different variations along the different

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axes of the global frame. Moreover, the value of the variation can vary significantly

from one CCP to another.

The proposed method is applied to piping assembly. As shown in section 4.1,

the actual pipe geometry depends on bending process capabilities, so the impact of

manufacturing deviations on tube geometry is easier simulated than with machining

process for instance. Nevertheless, the method can be displayed on other instances of

assembly since the inputs such as Stiffness Matrix, Key characteristics, process

capabilities, joints scheme and models of joints behaviour are defined. Moreover,

statistical tolerances with direct Monte Carlo simulation are employed, so

computational time directly depends on the number of elements used in the

simulations of manufacturing deviations.

Future extensions of this work include possibilities for adjustment solutions to

be implemented in the early design stage. The proposed method can be used to

compare different scenarii when multiple deviations occur. Last, the interaction of

different technologic choices at the design stage can be investigated.

Acknowledgements

This research work has been carried out in the frame of the GRC-Flexible Assembly of the

INNO’CAMPUS programme in partnership with EADS-IW.

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References

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CIRP Int. Sem. on Computer Aided

Tolerancing, 2001, France, 65-74.

Breteau, P., Thiebaut F., Bourdet P. and Falgarone H., Assembly simulation of

flexible parts through the fitting of linkage devices., Proc. of the 10th

CIRP Int.

Sem. on Computer Aided Tolerancing, 2007, Germany.

Camelio, J., Hu, S.J. and Ceglarek, D., Modeling Variation Propagation of Multi-

Station Assembly Systems with compliant Parts. Trans. of ASME J. of Mech.

Des., 2003, 125, 673-681.

Ceglarek, D.J. and Shi, J., Tolerance analysis for Sheet Metal Assembly using a

Beam-Based Model. Trans. of ASME Int. Mech. Eng. Cong. and Exp., 1997,

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Chang, M. and Gossard D.C., Modeling the assembly of compliant, non-ideal parts.

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Cid, G., Thiebaut, F., Bourdet, P. and Falgarone, H., Geometrical study of assembly

behaviour, taking into accounts rigid components’ deviations, actual geometric

variations and deformations. Proc. of the 9th

CIRP Int. Sem. on Computer Aided

Tolerancing, 2005, USA, ISBN 1-4020-5437-8, 301-310.

Dahlström, S., Lindkvist, L. and Söderberg, R., Practical Implications in Tolerance

Analysis of Sheet Metal Assemblies – Experiences from an Automotive

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CIRP Int. Sem. on Computer Aided Tolerancing,

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Modeling-Part II: A Generic 3D Variation Model for Rigid Body Assembly in

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CIRP Int. Sem. on Computer Aided Tolerancing, 2007,

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Points of interest

CCP 1 CCP 2 CCP 3 CCP 4 CCP 5 CCP 6 CCP 7 CCP 8

Mean µ

Standard deviation σ

6.13 N

3.98 N

3.95 N

2.13 N

3.23 N

2.06 N

2.18 N

1.25 N

2.32 N

1.25 N

3.59 N

1.99 N

5.43 N

2.91 N

11.04 N

6.03 N Forces

Yield Rate 83.14 % 99.02 % 99.37 % 100 % 100 % 99.48 % 92.47 % 77.86 %

Mean µ


0.279 mm

0.145 mm

0.542 mm

0.314 mm

0.356 mm

0.186 mm

0.357 mm

0.186 mm

0.357 mm

0.186 mm

0.357 mm

0.186 mm

0.354 mm

0.184 mm

0.330 mm

0.187 mm Displacements

Yield Rate 99.99 % 91 % 99.8 % 99.81 % 99.83 % 99.74 % 99.8 % 99.64 %

Table 1: Example of Global results of assembly sequence simulation

Outer diameter 15.87 mm

Thickness 0.81 mm

Material Titan alloy

Forming die 50 mm

Table 2: Geometrical and mechanical characteristics of case study

Table 3: associated LRA manufacturing coordinates to case study

CCP1 CCP2 CCP3 CCP4 CCP5 CCP6 CCP7 CCP8

X deviation 0.5 mm 0.5 mm 0.25 mm 0.25 mm 0.25 mm 0.25 mm 0.25 mm 0.25 mm

Y deviation 0.4 mm 0.4 mm 0.25 mm 0.25 mm 0.25 mm 0.25 mm 0.4 mm 0.4 mm

Z deviation 0.25 mm 0.25 mm 0.4 mm 0.4 mm 0.4 mm 0.4 mm 0.25 mm 0.25 mm

Table 4: Structure deviations on each CCP

L (mm) R (°) A (°)

54.85

58.64

164.93

182.57

87.14

1387.80

295.07

72.75

134.48

100.51

71.97

0

-5.01

180

5.01

0

26.41

-37.77

-25.46

-107.18

-15.64

0

90

13.62

13.62

40.67

49.33

50.14

46.56

48.28

25.05

35.61

0

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Table 5: Deviation of CCP position in the bracket frame

Step Sequence 1 Sequence 2 Sequence 3

1 Full tightening of CCP Installation of positioning CCP (i.e. 3, 6 & 7)

2 Installation of all CCP Installation of extremities

3 Tightening of extremities

Sequential positioning and tightening of remaining

CCP following the sequence 4

CCP 3 6 7 4 5 2 CCP 2 7 6 5 4 3

Table 6: Assembly sequences simulated

Points of interest


Mean µ


6.13 N

3.98 N

3.95 N

2.13 N

3.23 N

2.06 N

2.18 N

1.25 N

2.32 N

1.25 N

3.59 N

1.99 N

5.43 N

2.91 N

11.04 N

6.03 N Forces

Yield Rate 83.14 % 99.02 % 99.37 % 100 % 100 % 99.48 % 92.47 % 77.86 %

Mean µ


0.279 mm

0.145 mm

0.542 mm

0.314 mm

0.356 mm

0.186 mm

0.357 mm

0.186 mm

0.357 mm

0.186mm

0.357 mm

0.186 mm

0.354 mm

0.184 mm

0.330 mm


Yield Rate 99.99 % 91 % 99.8 % 99.81 % 99.83 % 99.74 % 99.8 % 99.64 %

Table 7: Global results of tolerance analysis with no assembly sequence simulation

Points of interest


Mean µ


7.00 N

4.68 N

4.53 N

2.55 N

3.28 N

2.13 N

2.24 N

1.27 N

2.42 N

1.31 N

3.84 N

2.12 N

5.38 N

2.88 N

9.78 N

5.53 N Forces

Rate of conformity 77.84 % 96.56 % 99.06 % 100 % 100 % 99.06 % 91.38 % 79.25 %

Mean µ


0.281 mm

0.146 mm

0.544 mm

0.321 mm

0.351 mm

0.183 mm

0.357 mm

0.188 mm

0.358 mm

0.188 mm

0.357 mm

0.184 mm

0.353 mm

0.183 mm

0.334 mm


Yield Rate 100 % 90.56 % 99.8 % 99.78 % 99.82 % 99.82 % 99.84 % 99.68 %

Table 8: Global results of tolerance analysis of assembly sequence 3 6 7 5 4 2

Points of interest


Mean µ


5.81 N

3.66 N

3.95 N

2.16 N

3.36 N

2.14 N

2.25 N

1.3 N

2.42 N

1.31 N

3.82 N

2.09 N

5.38 N

2.84 N

10.60 N

6.00 N Forces

Yield Rate 87.66 % 99.06 % 99.1 % 100 % 100 % 99.1 % 92.9 % 83 %

Mean µ


0.282 mm

0.147 mm

0.546 mm

0.320 mm

0.357 mm

0.187 mm

0.356 mm

0.185 mm

0.355 mm

0.186 mm

0.354 mm

0.184 mm

0.355 mm

0.183 mm

0.333 mm


Yield Rate 100 % 90.3 % 99.78 % 99.76 % 99.78 % 99.82 % 99.88 % 99.52 %

Table 9: Global results of tolerance analysis of assembly sequence 2 6 7 5 4 3

Xi direction 0.11 mm

Yi direction 0.11 mm

Zi direction 0.25 mm

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Figure 1: Geometrical KC variation contributors (Söderberg (1998))

Figure 2: Displacement representation of a compliant component

Figure 3 : classical 3-2-1 fixturing scheme

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Figure 4: Assembly of an actual tube on actual brackets

Figure 5a: synthetic graphical representation

Figure 5b: histogram representation

Figure 5: representation of force distribution

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Figure 6: Case Study

Figure 7: Bracket and structure frames

Figure 8: Manufacturing coordinates L, R, A

Figure 9: Intrinsic dL Bending process dispersion

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Figure 10: Local representations of the forces in CCP 1, CCP 2 and CCP 8 for

Tolerance analysis

Figure 11: Local representations of the forces in CCP1 and CCP8 and displacement in

CCP2 for assembly sequence 2 6 7 5 4 3

Figure 12: Local representations of the forces and displacements in CCP2 for

assembly sequence 2 6 7 5 4 3

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Assembly Sequence Influence on Geometric Deviations of